Question

If O is the origin and A is the point (a,b,c), then the equation of the plane through A and at right angles to OA is
A. $ a(x - a) - b(y - b) - c(z - c) = 0 $
B. $ a(x + a) + b(y + b) + c(z + c) = 0 $
C. $ a(x - a) + b(y - b) + c(z - c) = 0 $
D.None of the above

Answer 1

Hint: First of all we will take the normal plane equation in three dimensions and since it passes through the coordinates will place (a, b, c) and then will re-arrange for the resultant required answer.

Complete step-by-step answer:
Let us assume that a plane with normal $ ai + bj + ck $ can be given by the equation $ ax + by + cz + d = 0 $ ….. (A)
Since, the above equation passes through the points (a, b, c)
We get,
 $ a(a) + b(b) + c(c) + d = 0 $
Simplify the above equation-
 $ {a^2} + {b^2} + {c^2} + d = 0 $
Make the required term “d” the subject and move other terms on the opposite side. Remember when you move any term from one side to another, then the sign of the term also changes. Positive terms become negative and vice-versa.
 $ d = - ({a^2} + {b^2} + {c^2}) $
Place the above value in the equation (A)
 $ ax + by + cz - ({a^2} + {b^2} + {c^2}) = 0 $
Rearrange the above equation-
We get –
 $ a(x - a) + b(y - b) + c(z - c) = 0 $
This is the required solution.
From the given multiple choices – the option C is the correct answer.
So, the correct answer is “Option C”.

Note: Always remember the standard equation in three dimensions. Be good in basic mathematical simplification. Remember when you move any term from one side to another, then the sign of the terms also changes. Positive term becomes negative and negative term becomes positive.